https://r2.erweima.ai/mathbot/image/1720877032818lkiyd58o.png

To determine whether the given points \( P(-1, 2) \), \( Q(7, 5) \), and \( R(2, -6) \) form a right triangle and identify the hypotenuse, we need to calculate the distances between each pair of points. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Let's calculate the distances between each pair of points:

1. Distance between \(P(-1, 2)\) and \(Q(7, 5)\):
\[ PQ = \sqrt{(7 - (-1))^2 + (5 - 2)^2} = \sqrt{(7 + 1)^2 + 3^2} = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73} \]

2. Distance between \(Q(7, 5)\) and \(R(2, -6)\):
\[ QR = \sqrt{(2 - 7)^2 + (-6 - 5)^2} = \sqrt{(-5)^2 + (-11)^2} = \sqrt{25 + 121} = \sqrt{146} \]

3. Distance between \(R(2, -6)\) and \(P(-1, 2)\):
\[ PR = \sqrt{(2 - (-1))^2 + (-6 - 2)^2} = \sqrt{(2 + 1)^2 + (-8)^2} = \sqrt{3^2 + 64} = \sqrt{73} \]

We now have the distances:
\[ PQ = \sqrt{73} \]
\[ QR = \sqrt{146} \]
\[ PR = \sqrt{73} \]

Since \(PQ = PR\), the triangle is isosceles.

To check if the triangle is a right triangle, we need to verify if the Pythagorean theorem holds for any combination of these sides. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's check the possible right triangle configuration:
\[ (\sqrt{73})^2 + (\sqrt{73})^2 = (\sqrt{146})^2 \]
\[ 73 + 73 = 146 \]
\[ 146 = 146 \]

This shows that the triangle is indeed a right triangle. 

Since \(\sqrt{146}\) is the longest side, it must be the hypotenuse. Therefore, \(QR\) is the hypotenuse.

So, the triangle is both isosceles and right, and the correct answer is:
\[ \text{(d) Both (a) and (b)} \]

Would you like more details on this explanation or have any further questions?

Here are some additional questions you might be interested in:
1. How do I determine if three points form a triangle?
2. What are the properties of an isosceles triangle?
3. How can I find the area of a triangle given its vertices?
4. What are the different types of triangles?
5. How do I use the distance formula to find the length of a side in a coordinate plane?

**Tip:** When checking if a triangle is a right triangle using the Pythagorean theorem, always compare the square of the longest side with the sum of the squares of the other two sides.

Learn how to identify the hypotenuse in a right triangle formed by points in a coordinate plane using the distance formula and Pythagorean theorem. This problem covers key concepts in coordinate geometry and triangle properties, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation